nLab Chern class

Idea

The ordinary Chern classes are the integral characteristic classes

$$c_i : B U \to B^{2 i} \mathbb{Z}$$

of the classifying space $B U$ of the unitary group.

Accordingly these are characteristic classes in ordinary cohomology of U-principal bundles and hence of complex vector bundle

The first Chern class is the unique characteristic class of circle group-principal bundles.

The analogous classes for the orthogonal group are the Pontryagin classes.

More generally, there are generalized Chern classes for any complex oriented cohomology theory (Adams 74, Lurie 10).

Definition

Definition

For $n \geq 1$ the universal Chern classes

$$c_i \;\in\; H^{2i} \big( B U(n), \mathbb{Z} \big)$$

of the classifying space $B U(n)$ of the unitary group are the cohomology classes of $B U(n)$ in integral cohomology that are characterized as follows:

1. $c_0 = 1$ and $c_i = 0$ if $i \gt n$;

2. for $n = 1$, $c_1$ is the canonical generator of $H^2(B U(1), \mathbb{Z})\simeq \mathbb{Z}$;

3. under pullback along the inclusion $i : B U(n) \to B U(n+1)$ we have $i^* c_i^{(n+1)} = c_i^{(n)}$;

4. under the inclusion $B U(k) \times B U(l) \to B U(k+l)$ we have $i^* c_i = \sum_{j = 0}^i c_i \cup c_{j-i}$.

The corresponding total Chern class is the formal sum

$$c \;\coloneqq\; 1 + c_1 + c_2 + \cdots \;\in\; \underset{k}{\prod} H^{2k} \big( B U(n) \big)$$

Properties

Existence

Proposition

The cohomology ring of the classifying space $B U(n)$ (for the unitary group $U(n)$) is the polynomial ring on generators $\{c_k\}_{k = 1}^{n}$ of degree 2, called the Chern classes

$$H^\bullet(B U(n), \mathbb{Z}) \simeq \mathbb{Z}[c_1, \cdots, c_n] \,.$$

Moreover, for $B i \colon B U(n_1) \longrightarrow BU(n_2)$ the canonical inclusion for $n_1 \leq n_2 \in \mathbb{N}$, then the induced pullback map on cohomology

$$(B i)^\ast \;\colon\; H^\bullet(B U(n_2)) \longrightarrow H^\bullet(B U(n_1))$$

is given by

$$(B i)^\ast(c_k) \;=\; \left\{ \array{ c_k & for \; 1 \leq k \leq n_1 \\ 0 & otherwise } \right. \,.$$

(e.g. Kochman 96, theorem 2.3.1)

Proof

For $n = 1$, in which case $B U(1) \simeq \mathbb{C}P^\infty$ is the infinite complex projective space, we have (prop)

$$H^\bullet(B U(1)) \simeq \mathbb{Z}[ c_1 ] \,,$$

where $c_1$ is the first Chern class. From here we proceed by induction. So assume that the statement has been shown for $n-1$.

Observe that the canonical map $B U(n-1) \to B U(n)$ has as homotopy fiber the (2n-1)sphere (prop.) hence there is a homotopy fiber sequence of the form

$$S^{2n-1} \longrightarrow B U(n-1) \longrightarrow B U(n) \,.$$

Consider the induced Thom-Gysin sequence.

In odd degrees $2k+1 \lt 2n$ it gives the exact sequence

$$\cdots \to H^{2k}(B U(n-1)) \longrightarrow \underset{\simeq 0}{\underbrace{H^{2k+1-2n}(B U(n))}} \longrightarrow H^{2k+1}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}} \to \cdots \,,$$

where the right term vanishes by induction assumption, and the middle term since ordinary cohomology vanishes in negative degrees. Hence

$$H^{2k+1}(B U(n)) \simeq 0 \;\;\; for \; 2k+1 \lt 2n$$

Then for $2k+1 \gt 2n$ the Thom-Gysin sequence gives

$$\cdots \to H^{2k+1-2n}(B U(n)) \longrightarrow H^{2k+1}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}} \to \cdots \,,$$

where again the right term vanishes by the induction assumption. Hence exactness now gives that

$$H^{2k+1-2n}(B U(n)) \overset{}{\longrightarrow} H^{2k+1}(B U(n))$$

is an epimorphism, and so with the previous statement it follows that

$$H^{2k+1}(B U(n)) \simeq 0$$

for all $k$.

Next consider the Thom Gysin sequence in degrees $2k$

$$\cdots \to \underset{\simeq 0}{\underbrace{H^{2k-1}(B U(n-1))}} \longrightarrow H^{2k-2n}(B U(n)) \longrightarrow H^{2k}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} H^{2k}(B U(n-1)) \longrightarrow \underset{\simeq 0}{\underbrace{H^{2k +1 - 2n}(B U(n))}} \to \cdots \,.$$

Here the left term vanishes by the induction assumption, while the right term vanishes by the previous statement. Hence we have a short exact sequence

$$0 \to H^{2k-2n}(B U(n)) \longrightarrow H^{2k}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} H^{2k}(B U(n-1)) \to 0$$

for all $k$. In degrees $\bullet\leq 2n$ this says

$$0 \to \mathbb{Z} \overset{c_n \cup (-)}{\longrightarrow} H^{\bullet \leq 2n}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n} \to 0$$

for some Thom class $c_n \in H^{2n}(B U(n))$, which we identify with the next Chern class.

Since free abelian groups are projective objects in Ab, their extensions are all split (the Ext-group out of them vanishes), hence the above gives a direct sum decomposition

$$\begin{aligned} H^{\bullet \leq 2n}(B U(n)) & \simeq (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n} \oplus \mathbb{Z}\langle 2n\rangle \\ & \simeq (\mathbb{Z}[c_1, \cdots, c_{n}])_{\bullet \leq 2n} \end{aligned} \,.$$

Now by another induction over these short exact sequences, the claim follows.

First Chern class

• The first Chern class of a bundle $P$ is the class of its determinant line bundle $det P$

  $$c_1(P) = [det P] \,.$$

  See determinant line bundle for more.

Top Chern class

For $\mathcal{V}_X$ a complex vector bundle of complex rank $n$, the highest degree Chern class that may generally be non-vanishing is $c_n$. This is hence often called the top Chern class of the vector bundle.

Proposition

The top Chern class of a complex vector bundle $\mathcal{V}_X$ equals the Euler class $e$ of the underlying real vector bundle $\mathcal{V}^{\mathbb{R}}_X$:

$$\mathcal{V}_X \; \text{has complex rank}\;n \;\;\;\;\; \Rightarrow \;\;\;\;\; c_n \big( \mathcal{V}_X \big) \;\; = \;\; e \big( \mathcal{V}^{\mathbb{R}}_X \big) \;\;\;\; \in H^{2n} \big( X; \, \mathbb{Z} \big) \,.$$

(e.g. Bott-Tu 82 (20.10.6))

Proposition

The top Chern class of a complex vector bundle $\mathcal{V}_X$ equals the pullback of any Thom class $th \;\in\; H^{2n}\big( \mathcal{V}_X; \mathbb{Z} \big)$ on $\mathcal{V}_X$ along the zero-section:

$$\mathcal{V}_X \; \text{has complex rank}\;n \;\;\;\;\; \Rightarrow \;\;\;\;\; c_n \big( \mathcal{V}_X \big) \;\;\; = \;\;\; (0_X)^\ast (th) \;\; \in \; H^{2n} \big( X ; \, \mathbb{Z} \big)$$

(e.g. Bott-Tu 82, Prop. 12.4)

Splitting principle and Chern roots

Under the splitting principle all Chern classes are determined by first Chern classes:

Write $i \colon T \simeq U(1)^n \hookrightarrow U(n)$ for the maximal torus inside the unitary group, which is the subgroup of diagonal unitary matrices. Then

$$H^\bullet(B T, \mathbb{Z}) \simeq H^\bullet(B U(1)^n, \mathbb{Z})$$

is the polynomial ring in $n$ generators (to be thought of as the universal first Chern classes $c_i$ of each copy of $B U(1)$; equivalently as the weights of the group characters of $U(n)$) which are traditionally written $x_i$:

$$H^\bullet(B U(1)^n, \mathbb{Z}) \simeq \mathbb{Z}[x_1, \cdots, x_n] \,.$$

Write

$$B i \;\colon\; B U(1)^n \to B U(n)$$

for the induced map of deloopings/classifying spaces, then the $k$-universal Chern class $c_k \in H^{2k}(B U(n), \mathbb{Z})$ is uniquely characterized by the fact that its pullback to $B U(1)^n$ is the $k$th elementary symmetric polynomial $\sigma_k$ applied to these first Chern classes:

$$(B i)^\ast (c_k) = \sigma_k(x_1, \cdots, x_n) \,.$$

Equivalently, for $c = \sum_{i = 1}^n c_k$ the formal sum of all the Chern classes, and using the fact that the elementary symmetric polynomials $\sigma_k(x_1, \cdots, k_n)$ are the degree-$k$ piece in $(1+x_1) \cdots (1+x_n)$, this means that

$$(B i)^\ast (c) = (1+x_1) (1+ x_2) \cdots (1+ x_n) \,.$$

Since here on the right the first Chern classes $x_i$ appear as the roots of the Chern polynomial, they are also called Chern roots.

See also at splitting principle – Examples – Complex vector bundles and their Chern roots.

(e.g. Kochman 96, theorem 2.3.2, tom Dieck 08, theorem 19.3.2)

Lemma

For $n \in \mathbb{N}$ let $B \iota_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on ordinary cohomology

$$\left( B \iota_n \right) \;\colon\; H^\bullet( B U(n); \mathbb{Z} ) \longrightarrow H^\bullet( B U(1)^n; \mathbb{Z} )$$

is a monomorphism.

A proof of lemma , via analysis of the Serre spectral sequence of $U(n)/U(1)^n \to B U(1)^n \to B U(n)$ is indicated in (Kochman 96, p. 40). A proof via transfer of the Euler class of $U(n)/U(1)^n$, following (Dupont 78, (8.28)), is indicated at splitting principle (here).

Proposition

For $k \leq n \in \mathbb{N}$ let $B \iota_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on ordinary cohomology is of the form

$$(B i_n)^\ast \;\colon\; \mathbb{Z}[c_1, \cdots, c_k] \longrightarrow \mathbb{Z}[(c_1)_1,\cdots (c_1)_n]$$

and sends the $k$th Chern class $c_k$ (def. ) to the $k$th elementary symmetric polynomial in the $n$ copies of the first Chern class:

$$(B i_n)^\ast \;\colon\; c_k \mapsto \sigma_k( (c_1)_1, \cdots, (c_1)_n ) \,.$$

Proof

First consider the case $n = 1$.

The classifying space $B U(1)$ is equivalently the infinite complex projective space $\mathbb{C}P^\infty$. Its ordinary cohomology is the polynomial ring on a single generator $c_1$, the first Chern class (prop.)

$$H^\bullet(B U(1)) \simeq \mathbb{Z}[ c_1 ] \,.$$

Moreover, $B i_1$ is the identity and the statement follows.

Now by the Künneth theorem for ordinary cohomology (prop.) the cohomology of the Cartesian product of $n$ copies of $B U(1)$ is the polynomial ring in $n$ generators

$$H^\bullet(B U(1)^n) \simeq \mathbb{Z}[(c_1)_1, \cdots, (c_1)_n] \,.$$

By prop. the domain of $(B i_n)^\ast$ is the polynomial ring in the Chern classes $\{c_i\}$, and by the previous statement the codomain is the polynomial ring on $n$ copies of the first Chern class

$$(B i_n)^\ast \;\colon\; \mathbb{Z}[ c_1, \cdots, c_n ] \longrightarrow \mathbb{Z}[ (c_1)_1, \cdots, (c_1)_n ] \,.$$

This allows to compute $(B i_n)^\ast(c_k)$ by induction:

Consider $n \geq 2$ and assume that $(B i_{n-1})^\ast_{n-1}(c_k) = \sigma_k((c_1)_1, \cdots, (c_1)_{(n-1)})$. We need to show that then also $(B i_n)^\ast(c_k) = \sigma_k((c_1)_1,\cdots, (c_1)_n)$.

Consider then the commuting diagram

$$\array{ B U(1)^{n-1} &\overset{ B i_{n-1} }{\longrightarrow}& B U(n-1) \\ {}^{\mathllap{B j_{\hat t}}}\downarrow && \downarrow^{\mathrlap{B i_{\hat t}}} \\ B U(1)^n &\underset{B i_n}{\longrightarrow}& B U(n) }$$

where both vertical morphisms are induced from the inclusion

$$\mathbb{C}^{n-1} \hookrightarrow \mathbb{C}^n$$

which omits the $t$th coordinate.

Since two embeddings $i_{\hat t_1}, i_{\hat t_2} \colon U(n-1) \hookrightarrow U(n)$ differ by conjugation with an element in $U(n)$, hence by an inner automorphism, the maps $B i_{\hat t_1}$ and $B_{\hat i_{t_2}}$ are homotopic, and hence $(B i_{\hat t})^\ast = (B i_{\hat n})^\ast$, which is the morphism from prop. .

By that proposition, $(B i_{\hat t})^\ast$ is the identity on $c_{k \lt n}$ and hence by induction assumption

$$\begin{aligned} (B i_{n-1})^\ast (B i_{\hat t})^\ast c_{k \lt n} &= (B i_{n-1})^\ast c_{k \lt n} \\ = \sigma_k( (c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n ) \end{aligned} \,.$$

Since pullback along the left vertical morphism sends $(c_1)_t$ to zero and is the identity on the other generators, this shows that

$$(B i_n)^\ast(c_{k \lt n}) \simeq \sigma_{k\lt n}((c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n) \;\; mod (c_1)_t \,.$$

This implies the claim for $k \lt n$.

For the case $k = n$ the commutativity of the diagram and the fact that the right map is zero on $c_n$ by prop. shows that the element $(B j_{\hat t})^\ast (B i_n)^\ast c_n = 0$ for all $1 \leq t \leq n$. But by lemma the morphism $(B i_n)^\ast$, is injective, and hence $(B i_n)^\ast(c_n)$ is non-zero. Therefore for this to be annihilated by the morphisms that send $(c_1)_t$ to zero, for all $t$, the element must be proportional to all the $(c_1)_t$. By degree reasons this means that it has to be the product of all of them

$$\begin{aligned} (B i_n)^{\ast}(c_n) & = (c_1)_1 \otimes (c_1)_2 \otimes \cdots \otimes (c_1)_n \\ & = \sigma_n( (c_1)_1, \cdots, (c_1)_n ) \end{aligned} \,.$$

This completes the induction step.

Whitney sum formula

Proposition

For $k\leq n \in \mathbb{N}$, consider the canonical map

$$\mu_{k,n-k} \;\colon\; B U(k) \times B U(n-k) \longrightarrow B U(n)$$

(which classifies the Whitney sum of complex vector bundles of rank $k$ with those of rank $n-k$). Under pullback along this map the universal Chern classes (prop. ) are given by

$$(\mu_{k,n-k})^\ast(c_t) \;=\; \underoverset{i = 0}{t}{\sum} c_i \otimes c_{t-i} \,,$$

where we take $c_0 = 1$ and $c_j = 0 \in H^\bullet(B U(r))$ if $j \gt r$.

So in particular

$$(\mu_{k,n-k})^\ast(c_n) \;=\; c_k \otimes c_{n-k} \,.$$

e.g. (Kochman 96, corollary 2.3.4)

Proof

Consider the commuting diagram

$$\array{ H^\bullet( B U(n) ) &\overset{\mu_{k,n-k}^\ast}{\longrightarrow}& H^\bullet( B U(k) ) \otimes H^\bullet( B U(n-k) ) \\ {}^{\mathllap{\mu_k^\ast}}\downarrow && \downarrow^{\mathrlap{ \mu_{k}^\ast \otimes \mu_{n-k}^\ast }} \\ H^\bullet( B U(1)^n ) &\simeq& H^\bullet( B U(1)^k ) \otimes H^\bullet( B U(1)^{n-k} ) } \,.$$

This says that for all $t$ then

$$\begin{aligned} (\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t) & = \mu^\ast_n(c_t) \\ & = \sigma_t((c_1)_1, \cdots, (c_1)_n) \end{aligned} \,,$$

where the last equation is by prop. .

Now the elementary symmetric polynomial on the right decomposes as required by the left hand side of this equation as follows:

$$\sigma_t((c_1)_1, \cdots, (c_1)_n) \;=\; \underoverset{r = 0}{t}{\sum} \sigma_r((c_1)_1, \cdots, (c_1)_{n-k}) \cdot \sigma_{t-r}( (c_1)_{n-k+1}, \cdots, (c_1)_n ) \,,$$

where we agree with $\sigma_q((c_1)_1, \cdots, (c_1)_p) = 0$ if $q \gt p$. It follows that

$$(\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t) = (\mu_k^\ast \otimes \mu_{n-k}^\ast) \left( \underoverset{r=0}{t}{\sum} c_r \otimes c_{t-r} \right) \,.$$

Since $(\mu_k^\ast \otimes \mu_{n-k}^\ast)$ is a monomorphism by lemma , this implies the claim.

Chern-, Pontrjagin-, and Euler- characteristic forms

We spell out the formulas for the images under the Chern-Weil homomorphism of the Chern classes, Pontrjagin classes and Euler classes as characteristic forms over smooth manifolds.

Preliminaries

Let $X$ be a smooth manifold.

Write

(1)$$\Omega^{2\bullet}(X) \;\; \in \; CAlg_{\mathbb{R}}$$

for the commutative algebra over the real numbers of even-degree differential forms on $X$, under the wedge product of differential forms. This is naturally a graded commutative algebra, graded by form degree, but since we consider only forms in even degree it is actually a plain commutative algebra, too, after forgetting the grading.

Let $\mathfrak{g}$ be a semisimple Lie algebra (such as $\mathfrak{su}(d)$ or $\mathfrak{so}(d)$) with Lie algebra representation $V \,\in\, Rep_{\mathbb{C}}(\mathfrak{g})$ over the complex numbers of finite dimension $dim_{\mathbb{C}}(V) \,=\, n \,\in\, \mathbb{N}$ (for instance the adjoint representation or the fundamental representation), hence a homomorphism of Lie algebras

$$\mathfrak{g} \xrightarrow{\;\;\rho\;\;} End_{\mathbb{C}}(V)$$

to the linear endomorphism ring $End_{\mathbb{C}}(V)$, regarded here through its commutator as the endomorphism Lie algebra of $V$.

When regarded as an associative ring this is isomorphic to the matrix algebra of $n \times n$ square matrices

(2)$$End_{\mathbb{C}}(V) \;\; \simeq \;\; Mat_{n \times n}(\mathbb{C}) \,.$$

The tensor product of the $\mathbb{C}$-algebras (1) and (2)

is equivalently the $n \times n$ matrix algebra with coefficients in the complexification of even-degree differential forms:

$$\Omega^{2\bullet} \big(X\big) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V) \;\simeq\; \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \big( Mat_{n \times n}( \mathbb{R} ) \big) \;\; \simeq \;\; Mat_{n \times n} \big( \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \mathbb{C} \big) \,.$$

The multiplicative unit

(3)$$I \;\in\; Mat_{n \times n} \big( \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \mathbb{C} \big)$$

in this algebra is the smooth function (differential 0-forms) which is constant on the $n \times n$ identity matrix and independent of $t$.

Given a connection on a $G$-principal bundle, we regard its $\mathfrak{g}$-valued curvature form as an element of this algebra

(4)$$F_\nabla \,\in\, \Omega^2(X) \otimes_{\mathbb{R}} \mathfrak{g} \xrightarrow{\; \rho \;} \Omega^2(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V) \xhookrightarrow{\;\;\;} \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V)[t] \;\simeq\; Mat_{n \times n} \Big( \mathbb{C} \otimes_{\mathbb{R}} \Omega^{2}(X) \Big) \,.$$

The formulas

Chern forms

The total Chern form $c(\nabla)$ is the determinant of the sum of the unit (3) with the curvature form (4), and its component in degree $2k$, for $k \in \mathbb{N}$, is the $k$th Chern form $c_k(\nabla)$:

$$c(\nabla) \;\; \coloneqq \;\; \sum_k \underset{ \mathclap{ deg = 2k } }{ \underbrace{ c_k(\nabla) } } \;\; \coloneqq \;\; det \left( I + t \frac{i F_\nabla}{2\pi} \right) \,.$$

By the relation between determinant and trace, this is equal to the exponential of the trace of the logarithm of $I + \frac{i F_\nabla}{2\pi}$, this being the exponential series in the trace of the Mercator series in $\frac{i F_\nabla}{2\pi}$:

(5)$$\begin{aligned} c(\nabla) & \;=\; det \left( I + t \frac{i F_\nabla}{2\pi} \right) \\ & \;=\; \exp \circ tr \circ ln \left( I + \frac{i F_\nabla}{2\pi} \right) \\ & \;=\; \exp \circ tr \left( - \underset {k \in \mathbb{N}_+} {\sum} \tfrac{1}{k} \left( \frac{F_\nabla}{2\pi i} \right)^k \right) \\ & \;=\; \exp \left( \underset {k \in \mathbb{N}_+} {\sum} \tfrac{1}{k} \left( \frac { - (-i)^k } {(2\pi)^k} tr\big( F_\nabla^{\wedge_k} \big) \right) \right) \\ & \;=\; 1 \\ & \phantom{\;=\;} + \phantom{\frac{1}{1}} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right) \\ & \phantom{\;=\;} + \frac{1}{2} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right)^2 \\ & \phantom{\;=\;} + \frac{1}{6} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right)^3 \\ & \phantom{\;=\;} + \frac{1}{24} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right)^4 \\ & \phantom{\;=\;} + \cdots \\ & \;=\; 1 \\ & \phantom{\;=\;} + i \frac { tr\big(F_\nabla\big) } { 2 \pi } \\ & \phantom{\;=\;} + \tfrac{1}{2} \frac { tr\big( (F_\nabla)^2 \big) } { (2 \pi)^2 } + \frac{1}{2} \left( i \frac { tr\big( F_\nabla \big) } { 2\pi } \right)^2 \\ & \phantom{\;=\;} - i \tfrac{1}{3} \frac { tr\big( (F_\nabla)^3 \big) } { (2 \pi)^3 } + \frac{1}{2} \left( 2 \left( i \frac { tr\big( F_\nabla \big) } { 2 \pi } \right) \left( \tfrac{1}{2} \frac { tr\big( (F_\nabla)^2 \big) } { (2 \pi)^2 } \right) \right) + \frac{1}{6} \left( \left( i \frac { tr\big(F_\nabla\big) } { 2\pi } \right)^3 \right) \\ & \phantom{\;=\;} - \tfrac{1}{4} \frac {tr\big( (F_\nabla)^4 \big)} { (2 \pi)^4 } + \frac{1}{2} \left( \tfrac{1}{2} \frac {tr\big( (F_\nabla)^2 \big)} { (2 \pi)^2 } \right)^2 + \frac{1}{24} \left( i \frac {tr\big( F_\nabla \big)} { 2\pi } \right)^4 \\ & \phantom{\;=\;} + \cdots \\ & \;=\; 1 \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_1(\nabla) }{ \underbrace{ i \frac { tr\big(F_\nabla\big) } { 2 \pi } }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_2(\nabla) }{ \underbrace{ \frac {\tr\big( (F_\nabla)^2 \big) - \big( tr(F_\nabla) \big)^2 } { 8 \pi^2 } }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_3(\nabla) }{ \underbrace{ i \frac { - 2 \cdot tr\big( (F_\nabla)^3 \big) + 3 \cdot tr(F_\nabla) \cdot tr\big( (F_\nabla)^2 \big) - \big( tr(F_\nabla ) \big)^3 } {48 \pi^3} }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_4(\nabla) }{ \underbrace{ \frac { -6 \cdot tr\big( (F_\nabla)^4 \big) + 3 \cdot tr\big( (F_\nabla)^2 \big)^2 + \big( tr(F_\nabla) \big)^4 } {384 \pi^4} }} \\ & \phantom{\;=\;} + \cdots \end{aligned}$$

Pontrjagin forms

Setting $tr(F_\nabla) = 0$ in these expressions (5) yields the total Pontrjagin form $p(\nabla)$ with degree=$4k$-components the Pontrjagin forms $p_{k}(\nabla)$:

$$\begin{aligned} p(\nabla) & \;\coloneqq\; \underset{k \in \mathbb{N}}{\sum} \underset{ deg = 4k }{ \underbrace{ (-1)^{k} p_{k}(\nabla) } } \\ & \;=\; \underset{k \in \mathbb{N}}{\sum} \underset{ deg = 4k }{ \underbrace{ c_{2k}(\nabla) } } \\ & \;=\; 1 \\ & \phantom{\;=\;} + \underset{ \color{blue} = - p_1(\nabla) }{ \underbrace{ \frac {\tr\big( (F_\nabla)^2 \big) } { 8 \pi^2 } }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = p_2(\nabla) }{ \underbrace{ \frac { - 2 \cdot tr\big( (F_\nabla)^4 \big) + tr\big( (F_\nabla)^2 \big)^2 } {128 \pi^4} }} \\ \phantom{\;=\;} + \cdots \end{aligned}$$

Hence the first couple of Pontrjagin forms are

$$\begin{aligned} p_1(\nabla) & \;=\; - \frac {\tr\big( (F_\nabla)^2 \big) } { 8 \pi^2 } \\ p_2(\nabla) & \;=\; \frac { tr\big( (F_\nabla)^2 \big)^2 - 2 \cdot tr\big( (F_\nabla)^4 \big) } {128 \pi^4} \,. \end{aligned}$$

(See also, e.g., Nakahara 2003, Exp. 11.5)

Euler forms

For $n = 2k$ and with the curvature form again regarded as a 2-form valued $(2k) \times (2k)$-square matrix

$$F_{\nabla} \;=\; \big( (F_{\nabla})^a{}_b \big)_{1 \leq a,b, \leq 2k}$$

the Euler form is its Pfaffian of this matrix, hence the following sum over permutations $\sigma \in Sym(2k)$ with summands signed by the the signature $sgn(\sigma) \in \{\pm 1\}$:

$$\chi_{2k}(\nabla) \;=\; \frac {(-1)^k} { (4 \pi)^k \cdot k! } \underset{\sigma}{\sum} sgn(\sigma) \cdot (F_{\nabla})_{\sigma(1)\sigma(2)} \wedge (F_{\nabla})_{\sigma(3)\sigma(4)} \wedge \cdots \wedge (F_{\nabla})_{\sigma(2k-1)\sigma(2k)} \,.$$

The first of these is, using the Einstein summation convention and the Levi-Civita symbol:

$$\chi_4(\nabla) \;=\; \frac { \epsilon^{ a b c d} (F_{\nabla})_{a b} \wedge (F_\nabla)_{c d} } {32 \pi^2}$$

(See also, e.g., Nakahara 2003, Exp. 11.7)

Examples

Chern classes of linear representations

Under the Atiyah-Segal completion map linear representations of a group $G$ induce K-theory classes on the classifying space $B G$. Their Chern classes are hence invariants of the linear representations themselves.

See at characteristic class of a linear representation for more.

Related concepts

• Chern root

• Pontryagin class

• Stiefel-Whitney class

• Chern character

• Chern insulator

In Yang-Mills theory field configurations with non-vanishing second Chern class (and minimal energy) are called instantons. The second Chern class is the instanton number . For more on this see at SU(2)-instantons from the correct maths to the traditional physics story.

References

Original articles:

• Friedrich Hirzebruch, Chapter 1, Section 4 of: Neue topologische Methoden in der Algebraischen Geometrie, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 (doi:10.1007/978-3-662-41083-7)

• A. Grothendieck, La théorie des classes de Chern, Bulletin de la Société Mathématique de France 86 (1958), p. 137–154, (numdam:BSMF_1958__86__137_0)

Textbook accounts:

• Shoshichi Kobayashi, Katsumi Nomizu, Section XII.3 in: Foundations of Differential Geometry, Volume 1, Wiley 1963 (web, ISBN:9780471157335, Wikipedia)

• Werner Greub, Stephen Halperin, Ray Vanstone, chapter IX of volume II of Connections, Curvature, and Cohomology Academic Press (1973)

• John Milnor, James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)

• Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/BFb0063500)

• Stanley Kochman, section 2.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Tammo tom Dieck, Algebraic topology, EMS 2008

With an eye towards mathematical physics:

• Mikio Nakahara, Section 11.3 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

• Gerd Rudolph, Matthias Schmidt, Def. 4.2.2 of Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)

A brief introduction is in chapter 23, section 7

• Peter May, A concise course in algebraic topology (pdf)

For Conner-Floyd Chern classes in complex oriented cohomology theory:

• Frank Adams, part II.2 and part III.10 of Stable homotopy and generalised homology, 1974

• Jacob Lurie, Chromatic Homotopy Theory, 2010, lecture 4 (pdf) and lecture 5 (pdf)

See also

• Dupont, Curvature and characteristic classes, Springer 1978

• Timothy Hosgood, Chern classes of coherent analytic sheaves: a simplicial approach. Université d’Aix-Marseille (AMU), 2020. tel-02882140.

A MathOverflow question discussing references: https://mathoverflow.net/questions/345437/canonical-reference-for-chern-characteristic-classes